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×B is the notation often used to indicate the cross product with the vector B, a standard operation in three-dimensional vector mathematics and physics. When A × B is formed, A and B are vectors in three-dimensional space, and the result is a vector perpendicular to the plane containing A and B. The expression ×B can also denote the linear operator that maps any vector v to v × B.

Mathematically, the cross product with B can be represented by a matrix acting on v. If B = (B_x, B_y, B_z), the operator is given by the skew-symmetric matrix [B]_× = [[0, -B_z, B_y], [B_z, 0, -B_x], [-B_y, B_x, 0]]. Then v × B = [B]_× v. This matrix form makes the operator linear in v and highlights its skew-symmetric nature.

Key properties include bilinearity and antisymmetry: for vectors u, v and scalar α, (αu) × B = α(u × B) and u × (αB) = α(u × B), with u × B = −(B × u). The cross product is orthogonal to both operands, and its magnitude satisfies |u × B| = |u||B|sinθ, where θ is the angle between u and B. If u is parallel to B, then u × B = 0. In three dimensions, many identities involving cross products hold, though generalizations exist in higher dimensions.

Applications of ×B appear in physics and engineering, such as the Lorentz force F = q(v × B) and in describing rotational motion and gyroscopic effects. It also appears in vector calculus and electromagnetism, where cross products with field vectors are common.

Example: with B = (1, 2, 3) and v = (4, 5, 6), v × B = (3, −6, 3). See also cross product, skew-symmetric matrix, and Lorentz force.